Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 31

30

5

10 10

Time−delay
τ

Time (t), sec

(sec)

−5

−30

The signal x(t) = 7 e -t/2 cos(3t) and its autocorrelation function

Fig. 1.20

random noise). These properties of the correlation function are frequently used as
the basis for detecting signals in noise.

1.2.5 Signal Power and Energy

If x(t) is a signal, then its instantaneous power at any time t, is denoted by p(t), and
is defined as the power dissipated in a 1X-resistor when a voltage of amplitude
x(t) volts is applied. This power is given by the multiplication of the voltage and
the current as defined below:

p ð t Þ¼j x ð t Þj 2 ;

where absolute value is used to cater for the possibility of complex signals.
Since energy is defined as the integral of power w.r.t. time, the instantaneous
energy (e(t)), of the signal at any time instant t, is given by:
e ð t Þ¼ p ð t Þ dt ¼j x ð t Þj 2 dt ;
and the total energy E of the signal is given by:

Z

T = 2

j x ð t Þj 2 dt :

E ¼ lim
T !1

T = 2

The total average power is the time average of the total Energy, and is given by:

Z

T = 2

E
T ¼ lim

1
T

j x ð t Þj 2 dt : P ¼ lim

T !1

T !1

T = 2

Signals are classified as power signals, energy signals,orneither. Power signals
have finite power (hence, infinite energy, according to the above definitions).